Question

If a and b are related as $\sqrt{0.04\times 0.4\times a}=0.4\times 0.04\times \sqrt{b}$, then the value of $\dfrac{a}{b}$is
a) 0.016
b) 0.16
c) 1.60
d) None of these

Answer 1

Hint: We have the given equation: $\sqrt{0.04\times 0.4\times a}=0.4\times 0.04\times \sqrt{b}$. Since, both the variables are under root, we can square both sides. After squaring both sides of the equation, divide the whole equation by b and do the required calculations to get the value of $\dfrac{a}{b}$.

Complete step by step answer:
Since we have the expression:
$\sqrt{0.04\times 0.4\times a}=0.4\times 0.04\times \sqrt{b}......(1)$
Now, square both sides of the equation (1), we get:
\[{{\left( \sqrt{0.04\times 0.4\times a} \right)}^{2}}={{\left( 0.4\times 0.04\times \sqrt{b} \right)}^{2}}\]
\[0.04\times 0.4\times a=0.4\times 0.04\times 0.4\times 0.04\times b\]
$a=0.4\times 0.04\times b......(2)$
Now, divide equation (2) by b on both sides, we get:
$\begin{align}
  & \dfrac{a}{b}=0.4\times 0.04 \\
 & \dfrac{a}{b}=0.016 \\
\end{align}$
Hence $\dfrac{a}{b}=0.016$.

So, the correct answer is “Option A”.

Note: There is another way to solve the given expression. Instead of squaring both sides, we can write a number $x$ as $\sqrt{x}\times \sqrt{x}$.
So, for the given expression $\sqrt{0.04\times 0.4\times a}=0.4\times 0.04\times \sqrt{b}$, we can write:
\[\sqrt{0.04\times 0.4\times a}=\sqrt{0.4}\times \sqrt{0.4}\times \sqrt{0.04}\times \sqrt{0.04}\times \sqrt{b}\]
Now, we can cancel out the same terms on both sides of the equation. We get:
$\sqrt{a}=\sqrt{0.4\times 0.04\times b}$
Now, divide the whole equation by $\sqrt{b}$, we get:
$\sqrt{\dfrac{a}{b}}=\sqrt{0.4\times 0.04}$
Now, square both sides, we get:
$\begin{align}
  & \dfrac{a}{b}=0.4\times 0.04 \\
 & \dfrac{a}{b}=0.016 \\
\end{align}$